"hoarse," c.1722 in reference to a cattle disease (of persons, 1740), from husk on the notion of "dry as a husk." Earlier (1550s) "having husks." Sense of "tough and strong" (like corn husks) is first found 1869, American English. Related: Huskily; huskiness.

The moment any vessel is noticed steering for these islands [Whalefish Islands], the Esquimaux, or "Huskies,"* as the Danes customarily term them, come off in sufficient numbers to satisfy you that you are near the haunts of uncivilized men, and will afford sufficient information to guide any stranger to his anchorage. *"Husky" is their own term. I recollect the chorus to a song at Kamtchatka was "Husky, Husky." ["Last of the Arctic Voyages," London, 1855]